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Middle Years

4.09 Discriminant and parabolas (Extended)

Worksheet
Quadratic equations and parabolas
1

For the following graphs of functions of the form y = a x^{2} + b x + c :

i

State whether the vertex of the parabola is a maximum or minimum point.

ii

State whether the value of a is negative or positive.

iii

State the number of solutions to the equation a x^{2} + b x + c = 0.

a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
d
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
2

Below is the result after using the quadratic formula to solve an equation:

x = \dfrac{- \left( - 10 \right) \pm \sqrt{ - 1 }}{8}

What can be concluded about the solutions of the equation?

3

By inspection, determine the number of real solutions for each of the following quadratic equations:

a
x^{2} = 9
b
\left(x - 4\right)^{2} = 0
c
\left(x - 6\right)^{2} = - 2
The discriminant and equations
4

For the following quadratic equations:

i

Find the value of the discriminant.

ii
Hence state the number of real solutions for the equation.
a

x^{2} + 6 x + 9 = 0

b

4 x^{2} - 6 x + 7 = 0

c

2 x^{2} - 2 x = x - 1

d

x^{2} - 4 = 0

e
- x^{2} - 8 x - 16 = 0
f
- 2 x^{2} + 5 x - 6 = 0
g
- x^{2} + 6 x + 1 = 0
h
- x^{2} + 25 = 0
5

Consider the equation x^{2} + 22 x + 121 = 0.

a

Find the value of the discriminant.

b

State whether the solutions to the equation are rational or irrational.

6

For the following equations:

i

Find the value of the discriminant.

ii
State the nature of the roots.
a

5 x^{2} + 4 x + 8 = 0

b

4 x^{2} + 4 x - 6 = 0

c

4 x^{2} - 4 x + 1 = 0

d

x^{2} - 16 = 0

e

9x^{2} - 42 x = -49

f

2 x^{2} - 9 x = -2

g

x^2 = -6x -10

h

\dfrac{1}{4}x^{2} - 2 x + \dfrac{1}{4} = 0

7

Consider the equation x^{2} - 8 x - 48 = 0.

a

Find the discriminant.

b

Describe the nature of the roots.

c

Find the solutions of the equation.

8

For a particular quadratic equation b^{2} - 4 a c = 0, what can be said about the solutions of the quadratic equation?

9

The solutions of a quadratic equation are 9 and - 9. What can be said about the value of b^{2} - 4 a c?

10

Find an expression for the discriminant of the following quadratic equations:

a

m x^{2} + 3 x - 2 = 0

b

x^{2} - 4 n x - 2 = 0

c

x^{2} + 5 x + p - 5 = 0

d

-px^{2} + 4 x + p = 0

e

mx^{2} - 2m x + m - 3 = 0

f

\dfrac{3n}{4}x^{2} + \dfrac{1n}{2} x + 2n = 0

11

Find the range of values of a for which the quadratic equation a x^{2} - b x + c = 0, where b = 6 and c = 10, has two distinct real solutions.

12

Consider the equation in terms of x:

m x^{2} + 2 x - 1 = 0

Given that it has two unique solutions, determine the possible values of m.

13

Consider the equation in terms of x:

m x^{2} - 3 x - 5 = 0
a

Given that the equation has two unique solutions, determine the possible values of m.

b

State the value of m that must be eliminated from the range of solutions found in the previous part. Explain your answer.

14

Find the values of n for which x^{2} - 8 n x + 1296 = 0 has one solution.

15

Find the value of k for which 16 x^{2} + 8 x + k = 0 has equal roots.

16

Find the values of m for which \left(m + 4\right) x^{2} + 2 m x + 2 = 0 has a single solution.

17

For the following equations:

i

Find the discriminant in terms of k.

ii

Find the value of k so that the equation has equal solutions.

iii

Find the value of k so that the equation has real solutions.

iv

Find the value of k so that the equation has no real solutions.

v

Find the value of k so that the equation has real and distinct solutions.

a

2 x^{2} + 8 x + k = 0

b
k x^{2} + 20 x + 2 = 0
c

\left(k + 4\right) x^{2} + 10 x + 3 = 0

18

Consider the equation in terms of x:x^{2} + 18 x + k + 7 = 0

a

Find the values of k for which the equation has no real solutions.

b

State the smallest possible integer value of k.

The discriminant and parabolas
19

When graphing a particular parabola, Katrina used the quadratic formula and found that b^{2} - 4 a c = - 5. How many x-intercepts does the parabola have?

20

When graphing a particular parabola, Tony used the quadratic formula and found that \\ b^{2} - 4 a c = 0. How many x-intercepts does the parabola have?

21

For each of the following graphs of a quadratic f \left( x \right) = a x^{2} + b x + c, with discriminant \\ \Delta = b^{2} - 4ac :

i

State whether a \gt 0 or a \lt 0.

ii

State whether \Delta \gt 0, \Delta \lt 0 or \Delta = 0.

a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
d
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
e
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
f
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
g
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
h
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
i
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
j
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
k
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
l
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
m
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
n
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
o
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
p
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
22

Consider the graph of the quadratic function:y = m - 9 x - 3 x^{2}

a

Find the possible values of m, if the graph has no x-intercepts.

b

State the largest possible integer value of m.

23

Determine the value(s) of k for which the graph of y = 4 x^{2} - 4 x + k - 15 just touches the \\ x-axis.

Application
24

Consider a right-angled triangle with side lengths x units, x + p units and x + q units, ordered from shortest to longest. No two sides of this triangle have the same length.

a

Complete the statement:

p and q have lengths such that

0<⬚<⬚.

b

Write a quadratic equation that describes the relationship between the sides of the triangle in terms of x.

c

Find the discriminant of this quadratic equation.

d

State the number of real solutions to the quadratic equation. Explain your answer.

e

Find the value of x in terms of p when q = 2 p.

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